Folks we need to calm down and get behind the Prime Minister.
The Mail on Sunday said a "close ally" said it was "go-go-go" for Mr Johnson's leadership push.
Downing Street said Chancellor Philip Hammond, Foreign Secretary Boris Johnson, Home Secretary Amber Rudd, Brexit Secretary David Davis and Defence Secretary Sir Michael Fallon would remain in post.
"Clearly if she's got a worse result than two years ago and is nearly unable to form a government then she I doubt will survive in the long term as Conservative Party leader", he said.
The former Mayor of London, who was the chief cheerleader for Vote Leave during the European Union referendum campaign, has promised to "get on with the job".
The messages, seen by ITV News Political Correspondent Dan Hewitt, suggest that he is instead shoring up her support behind the scenes - at least for now.
British newspapers have been awash with speculation that Johnson will soon mount a leadership challenge after May gambled away a parliamentary majority in an election she did not need to call. We need somebody who can talk and connect with people like Jeremy Corbyn does. "Boris is the only option with the liberal values, Brexit credentials and popular appeal", he added.
"The prime minister has tonight spoken with the DUP to discus finalising a confidence and supply deal when parliament returns next week".